Vector Quantile Regression on Manifolds

Marco Pegoraro, Sanketh Vedula, Aviv A. Rosenberg, Irene Tallini, Emanuele Rodolà, Alexander Bronstein

Keywords: quantile regression, vector quantiles, optimal transport, directional statistics

TL;DR: We propose a novel approach for estimating high-dimensional conditional quantile functions on manifolds.

Abstract:

Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features.
QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain.
Although the notion of quantiles was recently extended to multi-variate distributions,
QR for multi-variate distributions on manifolds remains underexplored, even though
many important applications inherently involve data distributed on, e.g., spheres (climate measurements), tori (dihedral angles in proteins), or Lie groups (attitude in navigation).
By leveraging optimal transport theory and the notion of $c$-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs).
Our approach allows for quantile estimation, regression, and computation of conditional confidence sets.
We demonstrate the approach's efficacy and provide insights regarding the meaning of non-Euclidean quantiles through preliminary synthetic data experiments.

Submission Number: 81